Machine Kinematics
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The wraps add up to a turn

Every wrap angle in a closed run is computed on its own, from a pair of tangent lines that knows nothing about the others. Signed by which way the strand goes round, they add to exactly one turn — or to exactly nothing, for a crossed belt — and the integer is decided by the route rather than by any of the geometry.

Assumes Where a strand leaves a body.

15 min read 9 figures Two routes to the same numberExact and approximateSolved, not drawn

A closed strand is a closed curve, and a closed curve in the plane has a turning number: follow it all the way round and count how many complete turns the direction of travel makes. For a simple loop the answer is one. For a figure of eight it is nought. It is an integer, it cannot change while the curve is deformed without being cut or pushed through itself, and it is one of the oldest results about plane curves.

A taut strand turns nowhere except on the bodies it wraps. Along every straight run its direction is constant; at every body it turns by exactly the wrap angle, one way or the other. So the turning number of a run is the sum of its wrap angles, each signed by the sense of the wrap, divided by 2π2\pi — and that sum is an integer.

That sentence is doing more work than it looks. The wrap angles are computed independently, one body at a time, each from two tangent lines that were solved without reference to any other body in the run. There is nothing in the arithmetic that knows about the loop. Adding a dozen such angles and landing on 2π2\pi to fifteen decimal places is a statement about a quantity nothing in the computation was given.

The four routes, and the two integers

An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.
Fig. 1 Two pulleys wrapped the same way. The wraps are 191.478° and 168.522°, which is exactly 360°.

An open belt wraps both pulleys the same way, so both wraps are positive and they add to a full turn. That is why the classical formula has (π+2γ)(\pi + 2\gamma) and (π−2γ)(\pi - 2\gamma) in it: the two wraps are a half turn each, plus and minus the same correction. The correction is the angle the strand’s span makes with the line of centres, and it is the same angle at both ends because the span is one straight line.

A crossed belt — the arrangement an involute gear pair is — wraps the two pulleys opposite ways, so one wrap counts positive and the other negative, and the total is nought. That has an immediate consequence which is not obvious from any drawing: the two wrap angles of a crossed belt are equal. Not approximately, and not for equal pulleys — for any two radii, at any centre distance that admits the run. On the 40 and 24 mm pair 160 mm apart, both are 227.156357°.

A crossed belt, whose wraps cancel. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand crossed between them so the two turn opposite ways. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 227.16° and 227.16°, and they add to nothing at all, because the two are equal and of opposite sign — a crossed belt's path is a figure of eight and turns through zero. Length 547.0209 mm, against the textbook formula's 547.0209. positioned by solving, not by drawing.
Fig. 2 The same two pulleys, crossed. Both wraps are 227.156357°; the turning number has fallen from one to nought.

The proof is the conservation law read backwards. A crossed run has two wraps of opposite sign and they must sum to nought, so their magnitudes are equal — and that is the whole argument, with no reference to the radii at all. Written out the long way it is the observation that both wraps come to π+2γ\pi + 2\gamma with sin⁡γ=(r1+r2)/C\sin\gamma = (r_1 + r_2)/C, which is the form the textbook length formula takes and is usually presented as an algebraic coincidence of the derivation rather than as the thing it is.

An open belt gives the small pulley the smaller wrap — 168.5° against 191.5° here — and a crossed belt divides its grip evenly. Crossing a belt is usually described as a way of reversing the output. It is also a way of buying wrap on the pulley that has least of it, and the arithmetic above says exactly how much.

A loop over pegs — the case the hull identity is about — wraps all of them the same way, and the sum is still one turn however many there are. Three pegs, four, five, six, eight: the signed wraps come to 360.000000000° in every case, because the arcs of a taut loop are the exterior angles of the convex hull the loop is, and the exterior angles of any convex polygon add to a full turn.

A loop over four pegs is a hull and one circle. 4 equal pegs of radius 14 mm, with a strand pulled taut round them. The strand's straight runs are the edges of the hull of the centres, each pushed out by one radius, and its arcs are the hull's exterior angles. So the whole strand is 503.136 mm of hull perimeter plus 87.965 mm of one full circle — 591.1005 mm against the 591.1005 mm the run measures, and the count of pegs does not enter it. positioned by solving, not by drawing.
Fig. 3 Four pegs. Each arc is an exterior angle of the hull drawn underneath, and the four of them are one turn.

A run with an idler pressed against it from outside has one negative wrap among the positive ones, and the sum is one turn again.

What the negative wrap buys, and who pays for it

The last of those is the case with a design consequence, and the arithmetic hands it over directly.

If the total is fixed at one turn, then an idler that takes −25.342°-25.342° must be matched by +385.342°+385.342° of positive wrap elsewhere. A tensioner does not merely take up slack; it creates wrap, and exactly as much as it takes.

The interesting question is who receives it, and the answer is sharper than the bookkeeping requires. Swing the idler through its whole range of contact and watch all five wraps:

arm angle crank compressor pump alternator idler
4.123 109.089° 74.709° 80.359° 95.903° −0.061°
4.300 114.534° 74.709° 80.359° 104.091° −13.693°
4.450 117.685° 74.709° 80.359° 112.589° −25.342°
4.600 119.559° 74.709° 80.359° 124.219° −38.846°
4.774 120.399° 74.709° 80.359° 163.497° −78.964°

The two pulleys on either side of the idler gain all of it. The two on the far side of the loop — the compressor at 74.709° and the pump at 80.359° — do not move by so much as a thousandth of a degree, at any position of the arm.

That is worth stating as a rule, because it is the opposite of what “tensioning the belt” sounds like it should do. An idler buys wrap for its two neighbours and for nobody else. A tensioner placed on the slack span next to the crankshaft raises the crankshaft’s wrap and the wrap of whatever is on the other side of it, and does nothing at all for the alternator three pulleys away. Where to put the idler is therefore a question about which pulley is short of wrap, and it has a local answer.

What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing.
Fig. 4 The run’s length over the whole interval in which the idler is touching. The wrap it takes runs from 0.061° to 78.964°, and the same amount is handed to its two neighbours.

The other sum, which is not conserved at all

Signing the wraps is what makes them add to an integer, and it is worth asking what the unsigned sum does, because that one is a quantity a designer cares about.

Add the wrap angles without their signs and the serpentine above gives 410.684° at an arm angle of 4.450 — the four positive wraps plus the idler’s twenty-five degrees taken as a positive number. Do the same at the edge of contact and it gives 360.121°. In every case the total is exactly

360°+2×(the negative wrap),360° + 2 \times (\text{the negative wrap}),

which follows immediately from the conservation: the neighbours gain what the idler takes, and then the idler’s own arc is added again rather than subtracted.

So the two sums say different things and both are worth having. The signed one is an invariant and is a check, in the way a mobility count is. The unsigned one is a cost: it is how far the strand is bent, in total, on one lap of the mechanism. A belt that goes round a loop is bent through one turn no matter how many pulleys are in the loop or how they are arranged — that part is free — and every idler pressed against the outside of the run adds twice its own wrap to the bill, because it bends the strand the other way and then the neighbours bend it back.

An idler that barely touches does nothing. Move an idler by one millimetre along the bisector of its two spans and the run lengthens by 2 sin(θ/2), where θ is the wrap. The line is that closed form; the dots are the run's own length differentiated numerically, which knows nothing about the formula, and the two agree to 1.7e-7 across the whole interval of contact. The consequence is at the left-hand end: at a wrap of 0.06° the idler takes up 0.0011 mm per millimetre of its own travel. Authority vanishes exactly where contact does, so the useful part of a tensioner's range is strictly inside the part where it is touching.
Fig. 5 The same idler measured a different way. A tensioner set deep enough to take up 78.964° of wrap is bending the strand through 517.9° per lap where a loop with no idler at all bends it through 360°.

What that bending costs is not a question this field can answer — it needs a material, a thickness and a number of laps per minute — and the reason to compute it anyway is that the geometry half of a fatigue argument is exactly this sum, and it is the half that is usually guessed. A run with three idlers pressed in a little is often drawn as gentler than one with a single idler pressed in hard, and the arithmetic says whether it is: three at 20° cost 120° of extra bending and one at 50° costs 100°.

Which way round is one turn

One convention needs stating, because it decides the sign of everything above and is arbitrary.

A wrap counts positive when the strand goes round the body anticlockwise, which is the same as saying the body’s centre is on the strand’s left as it travels. Travelled the other way round, every wrap in a loop changes sign and the turning number of the open belt becomes −1-1 instead of +1+1. Nothing in the geometry cares; the length is the same and the wrap angles are the same; only the bookkeeping flips.

What does not depend on the convention is the difference between the routes. An open belt has ∣±1∣|{\pm}1| and a crossed one has 00, and no choice of direction turns one into the other. That is the sense in which the integer is a property of the route: it survives every deformation of the mechanism, and it survives reading the mechanism backwards.

The peg the strand never reaches. 5 equal pegs of radius 14 mm, with a strand pulled taut round them. The fifth peg is inside the convex hull of the other four, so the strand does not touch it: its wrap is not small, it is absent, and moving that peg anywhere inside the hull changes nothing about the length at all. So the whole strand is 503.136 mm of hull perimeter plus 87.965 mm of one full circle — 591.1005 mm against the 591.1005 mm the run measures, and the count of pegs does not enter it. positioned by solving, not by drawing.
Fig. 6 A fifth peg inside the hull. Its wrap is not zero-but-small; the strand does not touch it, and it contributes nothing to either sum.

There is a boundary case that makes the point. Slide a peg outward until it just touches the strand, and its wrap comes in at zero and grows continuously from there. Nothing jumps: a body that is just on the run contributes an arc of no length, no turning, and no wrap. So the integer is stable under a body joining or leaving the run, which is the property that makes it usable as a check on runs whose membership is decided by the geometry rather than declared.

The check this pays for

The reason to compute an integer that was known in advance is that it fails when something else is wrong, and in this field the thing that goes wrong is specific.

A tangent line exists between any two bodies that are not nested. So a run over a body the strand does not actually reach — an idler set just outside the span it was meant to press into — still returns tangent lines, still returns tangency points, and still returns a length. What it returns is a path that goes round that body the long way: the strand approaches, and instead of a wrap of six degrees it takes a wrap of three hundred and forty-eight, coming back round the far side to leave in the direction the next span needs.

Drawn, that path is a smooth closed curve. It crosses itself somewhere, but so does a crossed belt, and there is no reason a reader would look. It has a plausible length. Every wrap angle in it is a legitimate arc of a real circle. Nothing about the picture says anything is wrong.

Its turning number is two.

What the wraps add up to, and what decides it. Four runs this field draws, with every wrap angle signed by the way the strand goes round its body. The right-hand column is their sum divided by a full turn, and it is a whole number every time — the turning number of a closed plane curve, arrived at by adding up a handful of angles that were computed one at a time from tangent lines. It is 1 for a loop that goes round its pulleys once and 0 for a crossed belt, whose two wraps are equal and opposite whatever the two radii are. The serpentine's idler contributes -25.3°, and the total is still exactly one turn: a tensioner lengthens the path without changing what the path is.
Fig. 7 The four routes, with their sums. The right-hand column is the whole check: it is arithmetic on angles computed one at a time, and the answer is an integer that nothing in the computation was told.

That is how the case was found, and it is why the tautness test in this field’s library has three parts and puts the cheap one first: no straight run may pass inside a body, no two runs may cross, and the turning must be what the route says. The first two are geometry and cost a loop over every pair; the third is a sum that is already computed, and it is the one that fires.

Why the total cannot be measured wrong

The invariance also puts a floor under everything else in the field, and it is worth being precise about what kind of check it is.

It is not a comparison against a second implementation. There is no other route to the wrap angles here — they come from the tangency solve and nothing else — so an error in the tangency solve would move every wrap angle at once. What the turning number tests is whether those angles are consistent as a path: whether the direction the strand leaves each body is the direction it arrives at the next one, all the way round, with the bookkeeping closing.

Over forty randomly proportioned belts, open and crossed, the worst departure from the exact integer is 2×10−152 \times 10^{-15} radians. Over the whole contact range of the serpentine’s idler it is 9×10−169 \times 10^{-16}. Those are sums of five or six angles each of a radian or so, so they are running at the last bit of the double-precision arithmetic — which is what a quantity that is exactly right looks like when it is computed in floating point.

One routine, six strand systems. Every row is the same function: a list of bodies, each with a sense, handed to a routine that returns the tangent runs between them, the arcs on them, and the total. Nothing in it knows what a belt is, what a tackle is or what a tendon is. The right-hand column is what each row was checked against — a textbook formula, an integer, a convex hull's perimeter, a second route to the same length — and it is the reason the middle column can stay the same all the way down. positioned by solving, not by drawing.
Fig. 8 Two of the six rows in the field’s ledger are checked against integers rather than against formulas. They are the two cheapest checks and among the strongest.

What the integer does not settle

It says nothing about whether the run is short, whether it is sensible, or whether the mechanism would work.

A loop over four pegs and a loop over four pegs with an extra half turn thrown in around one of them are different objects with different lengths, and the second one has turning number two — so this test would catch it. But a strand routed round the outside of a pulley that should have been on the inside can have the right turning number and be an entirely different mechanism: the senses list is a genuine input to the problem, and no arithmetic recovers it. That is the subject of the essay on routes, and it is the one place in this field where a choice has to be made rather than computed.

Two taut paths, and no way between themA strand from one point to another past a peg. Each side gives a path that is taut — straight where it can be, on the surface where it must be, leaving at a right angle — and each is the shortest path on its own side: 219.165 mm on one and 200.643 mm on the other, against 200 mm of open air the strand cannot use. Neither can turn into the other without passing through the peg, which is a different kind of non-uniqueness from the assembly branches of a linkage: those are separate roots of one equation, and these are separate classes of one minimisation. positioned by solving, not by drawing.above 200.64 mm · below 219.17 mmstraight line 200 mm, unusable
Fig. 9 Two routes past one peg. Both are taut, both have the same turning, and they are 18.5 mm different in length.

Nor does the integer say anything about the distribution. A run whose total is one turn can put 340° of it on one pulley, which is a mechanism a tackle never has to worry about and five degrees on each of four others, and that is a real mechanism with a real problem — four pulleys that will not drive anything. The total is a conservation law, not a design.

A coarser invariant than the route

The integer is decided by the route and not by the geometry, and it is worth being exact about how much of the route it records, because that decides which routing errors the check catches.

A route is an element of a free group: which bodies the strand touches, in what order, and which side of each. The turning number is a single integer computed from it. So the map from routes to turning numbers is many-to-one, and the check sees only the image.

What it catches, therefore, is any error that changes the total. A wrap taken the wrong way round — a belt threaded crossed where it should be open, an idler on the wrong side of a run — changes a sign and moves the total by two, and the check fails immediately and loudly. That is the commonest routing mistake there is and it is exactly the one the integer is sensitive to.

What it misses is anything that preserves the total. Permuting the bodies leaves every wrap’s sign alone and reorders the runs, and the sum is unchanged — so a serpentine belt threaded round the right pulleys the right ways in the wrong order passes the check. So does a route with two errors that cancel: an idler on the wrong side and a pulley wrapped backwards, together, restore the integer and leave a mechanism that is wrong in two places.

That is not a defect in the check and it is worth saying why. An invariant that could distinguish every route would be the route, and computing the route is what the check is verifying rather than something it can assume. The value of a coarse invariant is precisely that it is computed by a different means from the thing it checks — the wraps come from pairs of tangent lines that know nothing about each other, and their sum comes out an integer anyway.

So the arrangement is the same one this site uses elsewhere: a cheap invariant that is sound in one direction, backed by something that decides the question. The turning number cannot certify a route and can refute one, the refutation is free, and the routes it fails to refute have to be checked another way — which for a strand means by the ordering, and the ordering is an input rather than a computation.

Which also says what a second check would have to look like. Not another integer, since every integer computed from the route is a coarser invariant again, but the ordered senses list itself, compared against what was intended. That is a comparison between two descriptions rather than a measurement, and it is the same terminus the topology field reaches when its own summaries run out.

What is not modelled

The turning number is a property of a closed curve, so it exists for a belt and not for a tendon, a tackle or a winch line, and the field’s routines refuse to compute one for an open run rather than returning nought. A strand that leaves the plane — a quarter-turn belt between shafts at right angles, which is a real and common arrangement — is outside this field entirely: the whole of the geometry above is planar, and the tangency condition in space is a different problem with a different answer. Nothing here counts a wrap that exceeds a full turn on one body, which is what a capstan or a winch drum does deliberately; the routine reports the wrap it finds and the arithmetic assumes each body is met once.

Where this goes next

Four tangents, and the two signs that choose between them. Two circles admit four common tangents, and a strand takes whichever one its two wrap senses name. Same sense at both ends — both centres on the same side of the strand — gives the two outer tangents, each 169.0444 mm long. Opposite senses give the two that cross between the circles, each 154.9193 mm. There is no search and no case analysis anywhere in this: the run's length is √(D² − Δ²) with Δ the signed radius difference, and changing one sense changes Δ from -18 to 70. positioned by solving, not by drawing. Where a strand leaves a body A taut strand meets the surface it lies on at a right angle, and every book draws it that way. It is not a rule about strands: it is what being shortest looks like, and a family of paths that were never told about tangency has its minimum exactly there — 200.64346 mm against the construction's 200.64346. Four pulleys, one idler, and one equation. A closed strand over four fixed pulleys and an idler carried on an arm. The idler sits outside the loop the other four make and bulges the strand out to reach it, so its wrap is signed the other way — -25.34° against the four positive ones — and the five still add to exactly one turn. The arm angle is not a free choice: the strand has a length, and matching it is one scalar equation, whatever the number of pulleys. At this position the run is 1018.046 mm and the idler takes up 0.4387 mm of strand for every millimetre it moves along the bisector of its two spans. positioned by solving, not by drawing. The tensioner is the unknown A linkage closes when a vector comes back to where it started: two equations, two unknowns. A strand closes when a number does — its length — however many bodies it runs over. So a run with one free body is determined, a run with two is not, and the arm angle that takes up 1,020 mm of belt is the root of one scalar equation solved to 1·10⁻¹³ mm. One routine, six strand systems. Every row is the same function: a list of bodies, each with a sense, handed to a routine that returns the tangent runs between them, the arcs on them, and the total. Nothing in it knows what a belt is, what a tackle is or what a tendon is. The right-hand column is what each row was checked against — a textbook formula, an integer, a convex hull's perimeter, a second route to the same length — and it is the reason the middle column can stay the same all the way down. positioned by solving, not by drawing. A member with no length of its own Every link of a pin-jointed machine holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a rope and a tendon do neither: a strand's path is decided by the bodies it touches, and its constraint is an inequality that does nothing at all until it is taut.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 10 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Belt driveConvex hullDesign ruleIdlerInvariantStrandTangencyTurning numberWrap angle